nLab
Sandbox

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Idea

Expositions of topological data analysis traditionally invoke point clouds – discrete subsets of some metric space – as the generic mathematical incarnation of datasets to be analyzed.

Maybe a more realistic and more encompassing model for sets of observed data – namely for measurement results – is the time-honored notion of a tuple of (values of) real observables, namely a continuous function

$$Obs \;\colon\; X \xrightarrow{\phantom{---}} \mathbb{R}^n$$

to the one-point compactification of $n$-dimensional Cartesian space.

A feature in the data as seen by such an observable is then an isosurface of $Obs$, hence the pre-image

$$Obs^{-1}(\vec v) \;\coloneqq\; \Big\{ x \in X \,\big\vert\, Obs(x) = \vec v \Big\} \;\subset\; X$$

of a tuple $\vec v \in \mathbb{R}^n$ of observed values.

Without restriction of generality we may assume that the observed value of interest is the origin $0 \in \mathbb{R}^n$, for if it is instead some $\vec v \in \mathbb{R}^n$ then we may instead pass to the observable $Obs \coloneqq Obs - const_{\vec v}$ without changing the essence of the situation.

In practice, the value of an observable can never be determined with the accuracy of a mathematical point in $\mathbb{R}^n$, instead there will be some positive real number $r \in \mathbb{R}_{\gt 0}$ such that one may hope (or wish) to measure $Obs$ up to measurement errors within a radius $r$. In this case, the desired isosurface could be any element in the set

$$\Big\{ f^{-1}(0) \subset X \,\vert\, f \in C^0(X,\mathbb{R}^n) ,\, \Vert f - Obs \Vert_\infty \lt r \Big\}$$

For example, $X$ might model the interior of a plasma container (say a fusion reactor) and $Ob = (T, p) : X \to \mathbb{R}^2$ could be the combined temperature- and pressure-observable (say as seen by laser probes into the plasma). Its isosurfaces are the intersections of given isobars with isotherms?.

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Illustrating the main theorem of persistent Cohomotopy.

Suppose we want to know the zeros of the observable $f$. If the resolution/error bar of measuring $f$ is $r$, then we know that

1. the zeros must be somewhere away from the gray region.

2. the homotopy class of $f$ on the quotient determines the number of zeros mod 2.